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Description
DifferentialSparseMultivariatePolynomial? implements an ordinary differential polynomial ring by combining a domain belonging to the category DifferentialVariableCategory? with the domain SparseMultivariatePolynomial?.
Author
William Sit
Date Created
19 July 1990
Date Last Updated
13 September 1991

Basic Operations

fricas
)show DifferentialPolynomialCategory
DifferentialPolynomialCategory(R: Ring,S: OrderedSet,V: DifferentialVariableCategory(t#2),E: OrderedAbelianMonoidSup)  is a category constructor
Abbreviation for DifferentialPolynomialCategory is DPOLCAT
This constructor is exposed in this frame.
------------------------------- Operations --------------------------------
?*? : (%,R) -> %                      ?*? : (R,%) -> %
?*? : (%,%) -> %                      ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> %        ?+? : (%,%) -> %
?-? : (%,%) -> %                      -? : % -> %
?=? : (%,%) -> Boolean                D : (%,(R -> R)) -> %
D : % -> % if R has DIFRING           D : (%,List(V)) -> %
D : (%,V) -> %                        1 : () -> %
0 : () -> %                           ?^? : (%,PositiveInteger) -> %
coefficient : (%,E) -> R              coefficients : % -> List(R)
coerce : S -> %                       coerce : V -> %
coerce : R -> %                       coerce : Integer -> %
coerce : % -> OutputForm              degree : % -> E
differentiate : (%,List(V)) -> %      differentiate : (%,V) -> %
eval : (%,List(V),List(%)) -> %       eval : (%,V,%) -> %
eval : (%,List(V),List(R)) -> %       eval : (%,V,R) -> %
eval : (%,%,%) -> %                   eval : (%,Equation(%)) -> %
ground : % -> R                       ground? : % -> Boolean
hash : % -> SingleInteger             initial : % -> %
isobaric? : % -> Boolean              latex : % -> String
leadingMonomial : % -> %              map : ((R -> R),%) -> %
mapExponents : ((E -> E),%) -> %      minimumDegree : % -> E
monomial : (R,E) -> %                 monomial? : % -> Boolean
monomials : % -> List(%)              one? : % -> Boolean
order : % -> NonNegativeInteger       pomopo! : (%,R,E,%) -> %
recip : % -> Union(%,"failed")        reductum : % -> %
retract : % -> S                      retract : % -> V
retract : % -> R                      sample : () -> %
separant : % -> %                     variables : % -> List(V)
weight : % -> NonNegativeInteger      zero? : % -> Boolean
?~=? : (%,%) -> Boolean
?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT))
?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT))
?*? : (NonNegativeInteger,%) -> %
?/? : (%,R) -> % if R has FIELD
D : (%,(R -> R),NonNegativeInteger) -> %
D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL)
D : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL)
D : (%,List(Symbol)) -> % if R has PDRING(SYMBOL)
D : (%,Symbol) -> % if R has PDRING(SYMBOL)
D : (%,NonNegativeInteger) -> % if R has DIFRING
D : (%,List(V),List(NonNegativeInteger)) -> %
D : (%,V,NonNegativeInteger) -> %
?^? : (%,NonNegativeInteger) -> %
associates? : (%,%) -> Boolean if R has ENTIRER or R has INTDOM and $has ATVCWC binomThmExpt : (%,%,NonNegativeInteger) -> % if$ has COMRING
characteristic : () -> NonNegativeInteger
charthRoot : % -> Union(%,"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) or R has CHARNZ coefficient : (%,V,NonNegativeInteger) -> % coefficient : (%,List(V),List(NonNegativeInteger)) -> % coerce : % -> % if R has GCDDOM or R has INTDOM and$ has ATVCWC
coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT))
conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) content : (%,V) -> % if R has GCDDOM content : % -> R if R has GCDDOM convert : % -> InputForm if V has KONVERT(INFORM) and R has KONVERT(INFORM) convert : % -> Pattern(Integer) if V has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) convert : % -> Pattern(Float) if V has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) degree : (%,S) -> NonNegativeInteger degree : (%,V) -> NonNegativeInteger degree : (%,List(V)) -> List(NonNegativeInteger) differentialVariables : % -> List(S) differentiate : (%,(R -> R)) -> % differentiate : (%,(R -> R),NonNegativeInteger) -> % differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) differentiate : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) differentiate : (%,Symbol) -> % if R has PDRING(SYMBOL) differentiate : (%,NonNegativeInteger) -> % if R has DIFRING differentiate : % -> % if R has DIFRING differentiate : (%,List(V),List(NonNegativeInteger)) -> % differentiate : (%,V,NonNegativeInteger) -> % discriminant : (%,V) -> % if R has COMRING eval : (%,List(S),List(R)) -> % if R has DIFRING eval : (%,S,R) -> % if R has DIFRING eval : (%,List(S),List(%)) -> % if R has DIFRING eval : (%,S,%) -> % if R has DIFRING eval : (%,List(%),List(%)) -> % eval : (%,List(Equation(%))) -> % exquo : (%,%) -> Union(%,"failed") if R has ENTIRER or R has INTDOM and$ has ATVCWC
exquo : (%,R) -> Union(%,"failed") if R has ENTIRER
factor : % -> Factored(%) if R has PFECAT
factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
gcd : (%,%) -> % if R has GCDDOM
gcd : List(%) -> % if R has GCDDOM
gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM
hashUpdate! : (HashState,%) -> HashState
isExpt : % -> Union(Record(var: V,exponent: NonNegativeInteger),"failed")
isPlus : % -> Union(List(%),"failed")
isTimes : % -> Union(List(%),"failed")
lcm : (%,%) -> % if R has GCDDOM
lcm : List(%) -> % if R has GCDDOM
lcmCoef : (%,%) -> Record(llcm_res: %,coeff1: %,coeff2: %) if R has GCDDOM
mainVariable : % -> Union(V,"failed")
makeVariable : % -> (NonNegativeInteger -> %) if R has DIFRING
makeVariable : S -> (NonNegativeInteger -> %)
minimumDegree : (%,List(V)) -> List(NonNegativeInteger)
minimumDegree : (%,V) -> NonNegativeInteger
monicDivide : (%,%,V) -> Record(quotient: %,remainder: %)
monomial : (%,V,NonNegativeInteger) -> %
monomial : (%,List(V),List(NonNegativeInteger)) -> %
multivariate : (SparseUnivariatePolynomial(%),V) -> %
multivariate : (SparseUnivariatePolynomial(R),V) -> %
numberOfMonomials : % -> NonNegativeInteger
order : (%,S) -> NonNegativeInteger
patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if V has PATMAB(INT) and R has PATMAB(INT)
patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if V has PATMAB(FLOAT) and R has PATMAB(FLOAT)
prime? : % -> Boolean if R has PFECAT
primitiveMonomials : % -> List(%)
primitivePart : (%,V) -> % if R has GCDDOM
primitivePart : % -> % if R has GCDDOM
reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT)
reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT)
reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R))
reducedSystem : Matrix(%) -> Matrix(R)
resultant : (%,%,V) -> % if R has COMRING
retract : % -> Integer if R has RETRACT(INT)
retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT))
retractIfCan : % -> Union(S,"failed")
retractIfCan : % -> Union(V,"failed")
retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT)
retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT))
retractIfCan : % -> Union(R,"failed")
smaller? : (%,%) -> Boolean if R has COMPAR
solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has PFECAT
squareFree : % -> Factored(%) if R has GCDDOM
squareFreePart : % -> % if R has GCDDOM
squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
subtractIfCan : (%,%) -> Union(%,"failed")
totalDegree : % -> NonNegativeInteger
totalDegree : (%,List(V)) -> NonNegativeInteger
totalDegreeSorted : (%,List(V)) -> NonNegativeInteger
unit? : % -> Boolean if R has ENTIRER or R has INTDOM and $has ATVCWC unitCanonical : % -> % if R has ENTIRER or R has INTDOM and$ has ATVCWC
unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has ENTIRER or R has INTDOM and $has ATVCWC univariate : % -> SparseUnivariatePolynomial(R) univariate : (%,V) -> SparseUnivariatePolynomial(%) weight : (%,S) -> NonNegativeInteger weights : (%,S) -> List(NonNegativeInteger) weights : % -> List(NonNegativeInteger) References http://en.wikipedia.org/wiki/Differential_algebra Kolchin, E.R. "Differential Algebra and Algebraic Groups" (Academic Press, 1973) Ritt, J.F. "Differential Algebra" (Dover, 1950). Example fricas odvar:=ODVAR Symbol (1) Type: Type fricas -- here are the first 5 derivatives of w -- the i-th derivative of w is printed as w subscript 5 [makeVariable('w,i)$odvar for i in 5..0 by -1] (2)
Type: List(OrderlyDifferentialVariable?(Symbol))
fricas
-- these are now algebraic indeterminates, ranked in an orderly way
-- in increasing order:
sort % (3)
Type: List(OrderlyDifferentialVariable?(Symbol))
fricas
-- we now make a general differential polynomial ring
-- instead of ODVAR, one can also use SDVAR for sequential ordering
dpol:=DSMP (FRAC INT, Symbol, odvar);
Type: Type
fricas
-- instead of using makeVariable, it is easier to
-- think of a differential variable w as a map, where
-- w.n is n-th derivative of w as an algebraic indeterminate
w := makeVariable('w)$dpol (4) Type: (NonNegativeInteger? -> DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))) fricas -- create another one called z, which is higher in rank than w -- since we are ordering by Symbol z := makeVariable('z)$dpol (5)
Type: (NonNegativeInteger? -> DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol)))
fricas
-- now define some differential polynomial
(f,b):dpol
Type: Void
fricas
f:=w.4::dpol - w.1 * w.1 * z.3 (6)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
b:=(z.1::dpol)^3 * (z.2)^2 - w.2 (7)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- compute the leading derivative appearing in b
lb:=leader b (8)
Type: OrderlyDifferentialVariable?(Symbol)
fricas
-- the separant is the partial derivative of b with respect to its leader
sb:=separant b (9)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- of course you can differentiate these differential polynomials
-- and try to reduce f modulo the differential ideal generated by b
-- first eliminate z.3 using the derivative of b
bprime:= differentiate b (10)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- find its leader
lbprime:= leader bprime (11)
Type: OrderlyDifferentialVariable?(Symbol)
fricas
-- differentiate f partially with respect to lbprime
pbf:=differentiate (f, lbprime) (12)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- to obtain the partial remainder of f with respect to b
ftilde:=sb * f- pbf * bprime (13)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- note high powers of lb still appears in ftilde
-- the initial is the leading coefficient when b is written
-- as a univariate polynomial in its leader
ib:=initial b (14)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- compute the leading coefficient of ftilde
-- as a polynomial in its leader
lcef:=leadingCoefficient univariate(ftilde, lb) (15)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))
fricas
-- now to continue eliminating the high powers of lb appearing in ftilde:
-- to obtain the remainder of f modulo b and its derivatives
f0:=ib * ftilde - lcef * b * lb (16)
Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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